Szabo's Conjecture A on equivariant Jiang-Su stability, in the unital stably finite case

Theorem 1.1: for $A$ simple, separable, unital, infinite-dimensional, nuclear, stably finite and $\mathcal Z$-stable, every action $\alpha$ of a countable discrete amenable group is cocycle conjugate to $\alpha\otimes\mathrm{id}_{\mathcal Z}$, with no restriction on how $\alpha$ acts on the trace simplex, and including finite groups and trivial actions. It does not treat nonunital algebras, and stably infinite (purely infinite) algebras are not part of this theorem.

Theorem 1.1: for $A$ simple, separable, unital, infinite-dimensional, nuclear, stably finite and $\mathcal Z$-stable, every action $\alpha$ of a countable discrete amenable group is cocycle conjugate to $\alpha\otimes\mathrm{id}_{\mathcal Z}$, with no restriction on how $\alpha$ acts on the trace simplex, and including finite groups and trivial actions. It does not treat nonunital algebras, and stably infinite (purely infinite) algebras are not part of this theorem.